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Theorem sbal1 2205
 Description: A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor . (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbal1
Distinct variable group:   ,
Allowed substitution hints:   (,,)

Proof of Theorem sbal1
StepHypRef Expression
1 sbequ12 1945 . . . . 5
21sps 1771 . . . 4
3 sbequ12 1945 . . . . . 6
43sps 1771 . . . . 5
54dral2 2056 . . . 4
62, 5bitr3d 248 . . 3
76a1d 24 . 2
8 nfa1 1807 . . . . . . . 8
98nfsb4 2131 . . . . . . 7
109nfrd 1780 . . . . . 6
11 sp 1764 . . . . . . . 8
1211sbimi 1665 . . . . . . 7
1312alimi 1569 . . . . . 6
1410, 13syl6 32 . . . . 5
1514adantl 454 . . . 4
16 sb4 2095 . . . . . . . 8
1716al2imi 1571 . . . . . . 7
1817hbnaes 2046 . . . . . 6
19 ax-7 1750 . . . . . 6
2018, 19syl6 32 . . . . 5
21 dveeq2 2078 . . . . . . . . 9
22 alim 1568 . . . . . . . . 9
2321, 22syl9 69 . . . . . . . 8
2423al2imi 1571 . . . . . . 7
25 sb2 2091 . . . . . . 7
2624, 25syl6 32 . . . . . 6
2726hbnaes 2046 . . . . 5
2820, 27sylan9 640 . . . 4
2915, 28impbid 185 . . 3
3029ex 425 . 2
317, 30pm2.61i 159 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wal 1550  wsb 1659 This theorem is referenced by:  sbal  2206 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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